- The Hamiltonian for several nuclei and electrons would be
where A runs over nuclei, i over electrons. Note that unlike an atom in free space, we now have a term in nuclear motion. Wave function would be

.

- Nuclei are
*much*heavier than electrons (at least 2000 times).*Decouple*problem: assert that there is no coupling between electornic motion (fast) and nuclear motion (slow). - Write wave function as
,
where we have a different electronic wave function for each value of the
parameters RA - coordinates of
*fixed*nuclei. Eigenfunction of - Electronic Schrödinger equation
the function E(RA) defines the

*potential energy surface*for nuclear motion. - Get rotational and vibrational energy levels for the nuclear part of the
problem, by solving
where is the nuclear motion wave function. Also determines chemical dynamics.

- Note that this is an approximation - properly speaking there are just
eigenvalues of the full Hamiltonian, representing "rovibronic" levels. But a
very good one.
- Fixing the nuclei has a drastic effect on the symmetry of the system!

- Even with the Born-Oppenheimer approimxation, we can make no progress on
analytical solution of the Schrödinger equation.
- Start simple: assume traditional molecular orbital ideas hold. For a system
of 2N electrons, assume tha the wave function is given by allocating electrons
in pairs to
*molecular orbitals*. - Physically, as we shall see, this is like assuming that the electrons interact
only with the
*average*potential of all the other electrons. Independent-particle model, or mean-field theory. - Assume that the MOs are orthonormal (and here real)
- Trial wave function is a Slater determinant
Substitute into the variation principle, and make stationary with respect to the form of the MOs.

- Need to evaluate
for this form of . Helped here by the orthonormality of the MOs and by the form of H.

- H contains zero-, one-, and two-electron operators only. In the integral over
all space, we have contributions from the coordinates of N-2 electrons that are
just products of integrals
so thanks to orthnormality we end up with only one- and two-electron integrals:

where

- The expression
is the energy for a given guess at the MOs. We optimize the MOs by minimizing this energy (variational method).

- Minimizing the energy requires satisfying the Hartree-Fock equations
is the

*Fock operator.* *Notes the sum over all k here. The solutions depend on themselves! This is not a simple eigenvalue problem.**Self-consistent field*(SCF) approach: guess a sert of MOs, construct**F**, diagonalize, and use its eigenvectors as a (hopefully) better guess at the MOs.

- For atoms, the symmetry of the system factors the SCF equations into an angular
part (spherical harmonics again!) and a radial part. The radial equations can
be solved numerically (one-dimensional problem).
- For molecules, the lower symmetry does not permit any factorization in general.
Numerical methods have been used for diatomic molecules, but becomes expensive
and cumbersome for polyatomic systems.
- How do we then express the MOs? Expand them in a fixed basis set, just like
the traditional LCAO (linear combination of atomic orbitals) reasoning used in
qualitative MO theory.

- Let us expand the unknown MOs linearly in a set of
*basis functions*:Here, can be any fucntions deemed appropriate. They are often approximations to atomic orbitals on the individual centres, and are often termed "atomic orbitals" (AOs).

- Note that the AOs are
*not*required to be orthonormal: - The SCF equations now become
where,

- Choose a basis
.
- Calculate the integrals over this basis (store on disk).
- Guess the MO coefficients
**C**. - Construct
**D**, then**F**. Solve the equations for a new**C**. If it does not agree with the previous estimate, iterate until it does.